Multivariate tail conditional expectation for scale mixtures of skew-normal distribution

In practice, a financial or actuarial data set may be a skewed or heavy-tailed and this motivates us to study a class of distribution functions in risk management theory that provide more information about these characteristics resulting in a more accurate risk analysis. In this paper, we consider a multivariate tail conditional expectation (MTCE) for multivariate scale mixtures of skew-normal (SMSN) distributions. This class of distributions contains skewed distributions and some members of this class can be used to analyse heavy-tailed data sets. We also provide a closed form for TCE in a univariate skew-normal distribution framework. Numerical examples are also provided for illustration.     

Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions

In this paper, we introduce a new risk measure, the so‐called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α‐quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, conditional value at risk or conditional tail variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where α ↓0 is no longer fixed). It is moreover assumed that the loss distribution is heavy tailed and depends on a covariate. The estimation method thus combines non‐parametric kernel methods with extreme‐value statistics. The asymptotic distribution of the estimators is established, and their finite‐sample behaviour is illustrated both on simulated data and on a real data set of daily rainfalls.     

Further remarks on asymptotic normality of likelihood and conditional analyses

Under weak conditions the normalized likelihood with or without weight function almost surely converges to a normal density function: for a real parameter or a vector parameter; with or without the assumption of independent identical distributions. Applications arise for confidence intervals, confidence distributions, structural distributions. and conditional analyses with transformation and structural models.     

On the characterization of generalized extreme value, power function, generalized Pareto and classical Pareto distributions by conditional expectation of record values

In the present paper, we give some theorems to characterize the generalized extreme value, power function, generalized Pareto (such as Pareto type II and exponential, etc.) and classical Pareto (Pareto type I) distributions based on conditional expectation of record values. Received: June 23, 1998; revised version: September 20, 1999     

Parameter estimation of nonlinear mixed-effects models using first-order conditional linearization and the EM algorithm

Nonlinear mixed-effects (NLME) models are flexible enough to handle repeated-measures data from various disciplines. In this article, we propose both maximum-likelihood and restricted maximum-likelihood estimations of NLME models using first-order conditional expansion (FOCE) and the expectation–maximization (EM) algorithm. The FOCE-EM algorithm implemented in the ForStat procedure SNLME is compared with the Lindstrom and Bates (LB) algorithm implemented in both the SAS macro NLINMIX and the S-Plus/R function nlme in terms of computational efficiency and statistical properties. Two realworld data sets an orange tree data set and a Chinese fir (Cunninghamia lanceolata) data set, and a simulated data set were used for evaluation. FOCE-EM converged for all mixed models derived from the base model in the two realworld cases, while LB did not, especially for the models in which random effects are simultaneously considered in several parameters to account for between-subject variation. However, both algorithms had identical estimated parameters and fit statistics for the converged models. We therefore recommend using FOCE-EM in NLME models, particularly when convergence is a concern in model selection.     

The exchange rate risk of Chinese yuan: Using VaR and ES based on extreme value theory

This paper applies extreme value theory (EVT) to estimate the tails of return series of Chinese yuan (CNY) exchange rates. We find that the degree of fitting Pareto distribution to the data of the tail of return series is extremely high. The empirical results indicate that expected shortfall cannot improve the tail risk problem of value-at-risk (VaR). The evidence of back testing indicates that EVT-based VaR values underestimate the risks of exchange rates such as USD/CNY and HKD/CNY, which may be caused by the continuous appreciation of CNY against USD and HKD. However, compared with VaR values calculated by historical simulation and variance–covariance method, VaR values calculated by EVT can measure the risk more accurately for the exchange rates of JPY/CNY and EUR/CNY.     

Joint modelling of longitudinal response and time-to-event data using conditional distributions: a Bayesian perspective

Over the last 20 or more years a lot of clinical applications and methodological development in the area of joint models of longitudinal and time-to-event outcomes have come up. In these studies, patients are followed until an event, such as death, occurs. In most of the work, using subject-specific random-effects as frailty, the dependency of these two processes has been established. In this article, we propose a new joint model that consists of a linear mixed-effects model for longitudinal data and an accelerated failure time model for the time-to-event data. These two sub-models are linked via a latent random process. This model will capture the dependency of the time-to-event on the longitudinal measurements more directly. Using standard priors, a Bayesian method has been developed for estimation. All computations are implemented using OpenBUGS. Our proposed method is evaluated by a simulation study, which compares the conditional model with a joint model with local independence by way of calibration. Data on Duchenne muscular dystrophy (DMD) syndrome and a set of data in AIDS patients have been analysed.     

On the estimation of the extreme value and normal distribution parameters based on progressive type-II hybrid-censored data

A progressive hybrid censoring scheme is a mixture of type-I and type-II progressive censoring schemes. In this paper, we mainly consider the analysis of progressive type-II hybrid-censored data when the lifetime distribution of the individual item is the normal and extreme value distributions. Since the maximum likelihood estimators (MLEs) of these parameters cannot be obtained in the closed form, we propose to use the expectation and maximization (EM) algorithm to compute the MLEs. Also, the Newton–Raphson method is used to estimate the model parameters. The asymptotic variance–covariance matrix of the MLEs under EM framework is obtained by Fisher information matrix using the missing information and asymptotic confidence intervals for the parameters are then constructed. This study will end up with comparing the two methods of estimation and the asymptotic confidence intervals of coverage probabilities corresponding to the missing information principle and the observed information matrix through a simulation study, illustrated examples and real data analysis.     

Skew generalized extreme value distribution: Probability-weighted moments estimation and application to block maxima procedure

ABSTRACT

Following the work of Azzalini (1985 Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Stat. 12:171–178.[Web of Science ®] , [Google Scholar] and 1986 Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica 46:199–208. [Google Scholar]) on the skew-normal distribution, we propose an extension of the generalized extreme value (GEV) distribution, the SGEV. This new distribution allows for a better fit of maxima and can be interpreted as both the distribution of maxima when maxima are taken on dependent data and when maxima are taken over a random block size. We propose to estimate the parameters of the SGEV distribution via the probability-weighted moment method. A simulation study is presented to provide an application of the SGEV on block maxima procedure and return level estimation. The proposed method is also implemented on a real-life data.     

Order statistics from extreme value distribution,i: tables of means,variances and covariances

Let X¹, X², …, Xⁿ be a random sample of size n from an extreme value distribution and X^1:n less than or equal X^2:n less than or equal … less than or equal X^n:n be the order statistics ob-tained from this sample. Tables of the means, variances, and covariances of the order statistics for samples of size n are given for n = 1(1)15(5)30. The computational formulae and procedure used and some checks employed are explained.     

Order statistics from extreme value distribution,ii: best linear unbiased estimates and some other uses

Let Y_r+1:n ≤ _Y:r+2:n ≤≤… <Y_n?6:n-<: TYPE-II censored sample from an extreme value population with µ and α as the location and scale parameters, respectively. Tables of coefficients for the best linear unbiased estimators (BLUEs) of µ and α are presented for various choices of censoring and sample sizes n = 2(1)15(5)30; variances and covariance of these estimators are also presented. The computational formulae and procedure used and some checks employed are explained. We finally illustrate some uses of the tables by taking examples.